As previously stated, dendrograms are unlabelled object which, in our context, may have a different number of leaves and do not hold any a-priori correspondence between the leaves in different objects.
The literature dealing with the comparison of dendrograms divide in two macro-areas, including (1) metrics defined for clustering dendrograms and (2) metrics designed for merge trees.
The first family of metrics mainly deals with labeled trees as byproducts of a hierarchical clustering algorithm. We refer to Flesia et al. [1]} for an exhaustive review of distance definitions. This kind of metrics are known to be heavily dependent on the graph structure of the dendrograms, leading to some limitations when comparing dendrograms with a different number of leaves. Moreover, theoretical continuity results with respect to dendrogram-associated point clouds are often lacking.
On the other hand, within topological data analysis, dendrograms are ofen referred as a particular case of merge trees, obtained when all the leaves of a merge tree lie at height 0. This allows to transfer merge tree literature, the second family of metrics, to dendrogram analysis.
In this Section we review the first family of metrics, detailing definitions and limitations of employing those distances in our context. Most of the metrics belonging to the second family, instead, shares one main drawback, namely the out of reach computational cost [2]}, [3]}, [4]}, which makes them unsuitable for our application.
Besides, the metrics with more performing algorithms [5]}, [6]} still lack the theoretical investigation to assess some practical properties, making them less worthy than others.
